Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $$\mathrm {SU}_{2,2}({\mathcal {O}}_K)$$ SU 2 , 2 ( O K ) where K is the imaginary-quadratic number field of discriminant $$-d$$ - d , $$d \in \{4, 7,8,11,15,19,20,24\}$$ d ∈ { 4 , 7 , 8 , 11 , 15 , 19 , 20 , 24 } we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

Two graded rings of Hermitian modular forms

We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in $${\mathbb {Q}}(\sqrt{-7})$$ Q ( - 7 ) and $${\mathbb {Q}}(\sqrt{-11})$$ Q ( - 11 ) . In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms.

Congruences in Hermitian Jacobi and Hermitian modular forms

In this paper, we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod p theory of Hermitian Jacobi forms over {\mathbb{Q}(i)}. We then apply the mod p theory of Hermitian Jacobi forms to characterize {U(p)} congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree 2 over {\mathbb{Q}(i)}.

A p-ADIC HERMITIAN MAASS LIFT

For K, an imaginary quadratic field with discriminant −DK, and associated quadratic Galois character χK, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level DK and nebentypus χK via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group U(2, 2) split over K. We generalize this (under certain conditions on K and p) to the case of p-oldforms of level pDK and character χK. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space of special Hermitian Jacobi forms. We then show how to adapt this construction to lift a Hida family of modular forms to a p-adic analytic family of automorphic forms in the Maass space of level p.